Book contents
- Frontmatter
- Introduction
- 1 The coefficient of resource utilization
- 2 A social equilibrium existence theorem
- 3 A classical tax-subsidy problem
- 4 Existence of an equilibrium for a competitive economy
- 5 Valuation equilibrium and Pareto optimum
- 6 Representation of a preference ordering by a numerical function
- 7 Market equilibrium
- 8 Economics under uncertainty
- 9 Topological methods in cardinal utility theory
- 10 New concepts and techniques for equilibrium analysis
- 11 A limit theorem on the core of an economy
- 12 Continuity properties of Paretian utility
- 13 Neighboring economic agents
- 14 Economies with a finite set of equilibria
- 15 Smooth preferences
- 16 Excess demand functions
- 17 The rate of convergence of the core of an economy
- 18 Four aspects of the mathematical theory of economic equilibrium
- 19 The application to economics of differential topology and global analysis
- 20 Least concave utility functions
20 - Least concave utility functions
Published online by Cambridge University Press: 05 January 2013
- Frontmatter
- Introduction
- 1 The coefficient of resource utilization
- 2 A social equilibrium existence theorem
- 3 A classical tax-subsidy problem
- 4 Existence of an equilibrium for a competitive economy
- 5 Valuation equilibrium and Pareto optimum
- 6 Representation of a preference ordering by a numerical function
- 7 Market equilibrium
- 8 Economics under uncertainty
- 9 Topological methods in cardinal utility theory
- 10 New concepts and techniques for equilibrium analysis
- 11 A limit theorem on the core of an economy
- 12 Continuity properties of Paretian utility
- 13 Neighboring economic agents
- 14 Economies with a finite set of equilibria
- 15 Smooth preferences
- 16 Excess demand functions
- 17 The rate of convergence of the core of an economy
- 18 Four aspects of the mathematical theory of economic equilibrium
- 19 The application to economics of differential topology and global analysis
- 20 Least concave utility functions
Summary
Introduction
The question of the representation of a convex preference preorder by a concave utility function was first raised and answered by de Finetti (1949), and further studied by Fenchel (1953, 1956), Moulin (1974), and by Kannai in the forthcoming article “Concavifiability and constructions of concave utility functions” which also discusses the problem of least concave utility functions. To illustrate the value of such a concave representation by one example, we consider an exchange economy ℰ whose consumers have convex preferences, and, following Scarf (1967), we associate with the economy ℰ a game without side payments in coalition form. If the preferences of each consumer are represented by a concave utility function, then the characteristic set of utility vectors of each coalition is convex, as in the original definition of Aumann-Peleg (1960). The convexity of these characteristic sets permits, for instance, a simplification [Scarf (1965) and Ekeland (1974); see also the related article of Shapley (1969)] of the proof of the non-emptiness of the core of Scarf (1967).
- Type
- Chapter
- Information
- Mathematical EconomicsTwenty Papers of Gerard Debreu, pp. 242 - 249Publisher: Cambridge University PressPrint publication year: 1983
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